In algebraic geometry, a sheaf of algebras on a
ringed space
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
''X'' is a
sheaf of commutative rings
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
on ''X'' that is also a
sheaf of -modules. It is
quasi-coherent
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
if it is so as a module.
When ''X'' is a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
, just like a ring, one can take the
global Spec of a quasi-coherent sheaf of algebras: this results in the contravariant functor
from the category of quasi-coherent (sheaves of)
-algebras on ''X'' to the category of schemes that are affine over ''X'' (defined below). Moreover, it is an equivalence: the quasi-inverse is given by sending an affine morphism
to
Affine morphism
A
morphism of schemes In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.
A morphism of algebraic stacks generalizes a ...
is called affine if
has an open affine cover
's such that
are affine. For example, a
finite morphism In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is ...
is affine. An affine morphism is
quasi-compact
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
and
separated; in particular, the direct image of a quasi-coherent sheaf along an affine morphism is quasi-coherent.
The base change of an affine morphism is affine.
Let
be an affine morphism between schemes and
a
locally ringed space
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
together with a map
. Then the natural map between the sets:
:
is bijective.
Examples
*Let
be the normalization of an algebraic variety ''X''. Then, since ''f'' is finite,
is quasi-coherent and
.
*Let
be a locally free sheaf of finite rank on a scheme ''X''. Then
is a quasi-coherent
-algebra and
is the associated vector bundle over ''X'' (called the total space of
.)
*More generally, if ''F'' is a coherent sheaf on ''X'', then one still has
, usually called the abelian hull of ''F''; see
Cone (algebraic geometry)#Examples.
The formation of direct images
Given a ringed space ''S'', there is the category
of pairs
consisting of a ringed space morphism
and an
-module
. Then the formation of direct images determines the contravariant functor from
to the category of pairs consisting of an
-algebra ''A'' and an ''A''-module ''M'' that sends each pair
to the pair
.
Now assume ''S'' is a scheme and then let
be the subcategory consisting of pairs
such that
is an affine morphism between schemes and
a quasi-coherent sheaf on
. Then the above functor determines the equivalence between
and the category of pairs
consisting of an
-algebra ''A'' and a quasi-coherent
-module
.
The above equivalence can be used (among other things) to do the following construction. As before, given a scheme ''S'', let ''A'' be a quasi-coherent
-algebra and then take its global Spec:
. Then, for each quasi-coherent ''A''-module ''M'', there is a corresponding quasi-coherent
-module
such that
called the sheaf associated to ''M''. Put in another way,
determines an equivalence between the category of quasi-coherent
-modules and the quasi-coherent
-modules.
See also
*
quasi-affine morphism
*
Serre's theorem on affineness
References
*
*{{Hartshorne AG
External links
*https://ncatlab.org/nlab/show/affine+morphism
Sheaf theory
Morphisms of schemes